EBay Inc. Chief Executive John Donahoe has battled to revive the online marketplace to take on other giants like Amazon.com Inc. Now, he's facing a very different foe: activist investor Carl Icahn.

Mr. Icahn wants to break up eBay by separating PayPal, its payments business, from the namesake shopping website, and he's nominated two business associates to the board of directors to help. More recently, he has called into question...